1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
|
(***************************************************************************)
(* This is part of aac_tactics, it is distributed under the terms of the *)
(* GNU Lesser General Public License version 3 *)
(* (see file LICENSE for more details) *)
(* *)
(* Copyright 2009-2010: Thomas Braibant, Damien Pous. *)
(***************************************************************************)
(** aac_rewrite -- rewriting modulo A or AC*)
open Ltac_plugin
module Control = struct
let debug = false
let printing = false
let time = false
end
module Debug = Helper.Debug (Control)
open Debug
let time_tac msg tac =
if Control.time then Coq.tclTIME msg tac else tac
let tac_or_exn tac exn msg = fun gl ->
try tac gl with e ->
let env = Tacmach.pf_env gl in
let sigma = Tacmach.project gl in
pr_constr env sigma "last goal" (Tacmach.pf_concl gl);
exn msg e
let retype = Coq.retype
open EConstr
open Names
open Proof_type
(** aac_lift : the ideal type beyond AAC_rewrite.v/Lift
A base relation r, together with an equivalence relation, and the
proof that the former lifts to the later. Howver, we have to
ensure manually the invariant : r.carrier == e.carrier, and that
lift connects the two things *)
type aac_lift =
{
r : Coq.Relation.t;
e : Coq.Equivalence.t;
lift : constr
}
type rewinfo =
{
hypinfo : Coq.Rewrite.hypinfo;
in_left : bool; (** are we rewriting in the left hand-sie of the goal *)
pattern : constr;
subject : constr;
morph_rlt : Coq.Relation.t; (** the relation we look for in morphism *)
eqt : Coq.Equivalence.t; (** the equivalence we use as workbase *)
rlt : Coq.Relation.t; (** the relation in the goal *)
lifting: aac_lift
}
let infer_lifting (rlt: Coq.Relation.t) (k : lift:aac_lift -> Proof_type.tactic) : Proof_type.tactic =
Coq.cps_evar_relation rlt.Coq.Relation.carrier (fun e ->
let lift_ty =
mkApp (Lazy.force Theory.Stubs.lift,
[|
rlt.Coq.Relation.carrier;
rlt.Coq.Relation.r;
e
|]
) in
Coq.cps_resolve_one_typeclass ~error:(Pp.strbrk "Cannot infer a lifting")
lift_ty (fun lift goal ->
let x = rlt.Coq.Relation.carrier in
let r = rlt.Coq.Relation.r in
let eq = (Coq.nf_evar goal e) in
let equiv = Coq.lapp Theory.Stubs.lift_proj_equivalence [| x;r;eq; lift |] in
let lift =
{
r = rlt;
e = Coq.Equivalence.make x eq equiv;
lift = lift;
}
in
k ~lift:lift goal
))
(** Builds a rewinfo, once and for all *)
let dispatch in_left (left,right,rlt) hypinfo (k: rewinfo -> Proof_type.tactic ) : Proof_type.tactic=
let l2r = hypinfo.Coq.Rewrite.l2r in
infer_lifting rlt
(fun ~lift ->
let eq = lift.e in
k {
hypinfo = hypinfo;
in_left = in_left;
pattern = if l2r then hypinfo.Coq.Rewrite.left else hypinfo.Coq.Rewrite.right;
subject = if in_left then left else right;
morph_rlt = Coq.Equivalence.to_relation eq;
eqt = eq;
lifting = lift;
rlt = rlt
}
)
(** {1 Tactics} *)
(** Build the reifiers, the reified terms, and the evaluation fonction *)
let handle eqt zero envs (t : Matcher.Terms.t) (t' : Matcher.Terms.t) k =
let (x,r,_) = Coq.Equivalence.split eqt in
Theory.Trans.mk_reifier (Coq.Equivalence.to_relation eqt) zero envs
(fun (maps, reifier) ->
(* fold through a term and reify *)
let t = Theory.Trans.reif_constr_of_t reifier t in
let t' = Theory.Trans.reif_constr_of_t reifier t' in
(* Some letins *)
let eval = (mkApp (Lazy.force Theory.Stubs.eval, [|x;r; maps.Theory.Trans.env_sym; maps.Theory.Trans.env_bin; maps.Theory.Trans.env_units|])) in
Coq.cps_mk_letin "eval" eval (fun eval ->
Coq.cps_mk_letin "left" t (fun t ->
Coq.cps_mk_letin "right" t' (fun t' ->
k maps eval t t'))))
(** [by_aac_reflexivity] is a sub-tactic that closes a sub-goal that
is merely a proof of equality of two terms modulo AAC *)
let by_aac_reflexivity zero
eqt envs (t : Matcher.Terms.t) (t' : Matcher.Terms.t) : Proof_type.tactic =
handle eqt zero envs t t'
(fun maps eval t t' ->
let (x,r,e) = Coq.Equivalence.split eqt in
let decision_thm = Coq.lapp Theory.Stubs.decide_thm
[|x;r;e;
maps.Theory.Trans.env_sym;
maps.Theory.Trans.env_bin;
maps.Theory.Trans.env_units;
t;t';
|]
in
(* This convert is required to deal with evars in a proper
way *)
let convert_to = mkApp (r, [| mkApp (eval,[| t |]); mkApp (eval, [|t'|])|]) in
let convert = Proofview.V82.of_tactic (Tactics.convert_concl convert_to Constr.VMcast) in
let apply_tac = Proofview.V82.of_tactic (Tactics.apply decision_thm) in
(Tacticals.tclTHENLIST
[ retype decision_thm; retype convert_to;
convert ;
tac_or_exn apply_tac Coq.user_error (Pp.strbrk "unification failure");
tac_or_exn (time_tac "vm_norm" (Proofview.V82.of_tactic (Tactics.normalise_in_concl))) Coq.anomaly "vm_compute failure";
Tacticals.tclORELSE (Proofview.V82.of_tactic Tactics.reflexivity)
(Tacticals.tclFAIL 0 (Pp.str "Not an equality modulo A/AC"))
])
)
let by_aac_normalise zero lift ir t t' =
let eqt = lift.e in
let rlt = lift.r in
handle eqt zero ir t t'
(fun maps eval t t' ->
let (x,r,e) = Coq.Equivalence.split eqt in
let normalise_thm = Coq.lapp Theory.Stubs.lift_normalise_thm
[|x;r;e;
maps.Theory.Trans.env_sym;
maps.Theory.Trans.env_bin;
maps.Theory.Trans.env_units;
rlt.Coq.Relation.r;
lift.lift;
t;t';
|]
in
(* This convert is required to deal with evars in a proper
way *)
let convert_to = mkApp (rlt.Coq.Relation.r, [| mkApp (eval,[| t |]); mkApp (eval, [|t'|])|]) in
let convert = Proofview.V82.of_tactic (Tactics.convert_concl convert_to Constr.VMcast) in
let apply_tac = Proofview.V82.of_tactic (Tactics.apply normalise_thm) in
(Tacticals.tclTHENLIST
[ retype normalise_thm; retype convert_to;
convert ;
apply_tac;
])
)
(** A handler tactic, that reifies the goal, and infer the liftings,
and then call its continuation *)
let aac_conclude
(k : constr -> aac_lift -> Theory.Trans.ir -> Matcher.Terms.t -> Matcher.Terms.t -> Proof_type.tactic) = fun goal ->
let (equation : types) = Tacmach.pf_concl goal in
let envs = Theory.Trans.empty_envs () in
let left, right,r =
match Coq.match_as_equation goal equation with
| None -> Coq.user_error @@ Pp.strbrk "The goal is not an applied relation"
| Some x -> x in
try infer_lifting r
(fun ~lift goal ->
let eq = Coq.Equivalence.to_relation lift.e in
let tleft,tright, goal = Theory.Trans.t_of_constr goal eq envs (left,right) in
let goal, ir = Theory.Trans.ir_of_envs goal eq envs in
let concl = Tacmach.pf_concl goal in
let env = Tacmach.pf_env goal in
let sigma = Tacmach.project goal in
let _ = pr_constr env sigma "concl "concl in
let evar_map = Tacmach.project goal in
Tacticals.tclTHEN
(Refiner.tclEVARS evar_map)
(k left lift ir tleft tright)
goal
)goal
with
| Not_found -> Coq.user_error @@ Pp.strbrk "No lifting from the goal's relation to an equivalence"
open Tacexpr
let aac_normalise = fun goal ->
let ids = Tacmach.pf_ids_of_hyps goal in
let mp = MPfile (DirPath.make (List.map Id.of_string ["AAC"; "AAC_tactics"])) in
let norm_tac = KerName.make2 mp (Label.make "internal_normalize") in
let norm_tac = Locus.ArgArg (None, norm_tac) in
Tacticals.tclTHENLIST
[
aac_conclude by_aac_normalise;
Proofview.V82.of_tactic (Tacinterp.eval_tactic (TacArg (None, TacCall (None, (norm_tac, [])))));
Proofview.V82.of_tactic (Tactics.keep ids)
] goal
let aac_reflexivity = fun goal ->
aac_conclude
(fun zero lift ir t t' ->
let x,r = Coq.Relation.split (lift.r) in
let r_reflexive = (Coq.Classes.mk_reflexive x r) in
Coq.cps_resolve_one_typeclass ~error:(Pp.strbrk "The goal's relation is not reflexive")
r_reflexive
(fun reflexive ->
let lift_reflexivity =
mkApp (Lazy.force (Theory.Stubs.lift_reflexivity),
[|
x;
r;
lift.e.Coq.Equivalence.eq;
lift.lift;
reflexive
|])
in
Tacticals.tclTHEN
(Tacticals.tclTHEN (retype lift_reflexivity) (Proofview.V82.of_tactic (Tactics.apply lift_reflexivity)))
(fun goal ->
let concl = Tacmach.pf_concl goal in
let env = Tacmach.pf_env goal in
let sigma = Tacmach.project goal in
let _ = pr_constr env sigma "concl "concl in
by_aac_reflexivity zero lift.e ir t t' goal)
)
) goal
(** A sub-tactic to lift the rewriting using Lift *)
let lift_transitivity in_left (step:constr) preorder lifting (using_eq : Coq.Equivalence.t): tactic =
fun goal ->
(* catch the equation and the two members*)
let concl = Tacmach.pf_concl goal in
let (left, right, _ ) = match Coq.match_as_equation goal concl with
| Some x -> x
| None -> Coq.user_error @@ Pp.strbrk "The goal is not an equation"
in
let lift_transitivity =
let thm =
if in_left
then
Lazy.force Theory.Stubs.lift_transitivity_left
else
Lazy.force Theory.Stubs.lift_transitivity_right
in
mkApp (thm,
[|
preorder.Coq.Relation.carrier;
preorder.Coq.Relation.r;
using_eq.Coq.Equivalence.eq;
lifting;
step;
left;
right;
|])
in
Tacticals.tclTHENLIST
[ retype lift_transitivity;
Proofview.V82.of_tactic (Tactics.apply lift_transitivity)
] goal
(** The core tactic for aac_rewrite. Env and sigma are for the constr *)
let core_aac_rewrite ?abort
rewinfo
subst
(by_aac_reflexivity : Matcher.Terms.t -> Matcher.Terms.t -> Proof_type.tactic)
env sigma (tr : constr) t left : tactic =
pr_constr env sigma "transitivity through" tr;
let tran_tac =
lift_transitivity rewinfo.in_left tr rewinfo.rlt rewinfo.lifting.lift rewinfo.eqt
in
Coq.Rewrite.rewrite ?abort rewinfo.hypinfo subst (fun rew ->
Tacticals.tclTHENSV
(tac_or_exn (tran_tac) Coq.anomaly "Unable to make the transitivity step")
(
if rewinfo.in_left
then [| by_aac_reflexivity left t ; rew |]
else [| by_aac_reflexivity t left ; rew |]
)
)
exception NoSolutions
(** Choose a substitution from a
[(int * Terms.t * Env.env Search_monad.m) Search_monad.m ] *)
(* WARNING: Beware, since the printing function can change the order of the
printed monad, this function has to be updated accordingly *)
let choose_subst subterm sol m=
try
let (depth,pat,envm) = match subterm with
| None -> (* first solution *)
List.nth ( List.rev (Search_monad.to_list m)) 0
| Some x ->
List.nth ( List.rev (Search_monad.to_list m)) x
in
let env = match sol with
None ->
List.nth ( (Search_monad.to_list envm)) 0
| Some x -> List.nth ( (Search_monad.to_list envm)) x
in
pat, env
with
| _ -> raise NoSolutions
(** rewrite the constr modulo AC from left to right in the left member
of the goal *)
let aac_rewrite_wrap ?abort rew ?(l2r=true) ?(show = false) ?(in_left=true) ?strict ~occ_subterm ~occ_sol ?extra : Proof_type.tactic = fun goal ->
let envs = Theory.Trans.empty_envs () in
let (concl : types) = Tacmach.pf_concl goal in
let (_,_,rlt) as concl =
match Coq.match_as_equation goal concl with
| None -> Coq.user_error @@ Pp.strbrk "The goal is not an applied relation"
| Some (left, right, rlt) -> left,right,rlt
in
let check_type x =
Tacmach.pf_conv_x goal x rlt.Coq.Relation.carrier
in
Coq.Rewrite.get_hypinfo rew ~l2r ?check_type:(Some check_type)
(fun hypinfo ->
dispatch in_left concl hypinfo
(
fun rewinfo ->
let goal =
match extra with
| Some t -> Theory.Trans.add_symbol goal rewinfo.morph_rlt envs (EConstr.to_constr (Tacmach.project goal) t)
| None -> goal
in
let pattern, subject, goal =
Theory.Trans.t_of_constr goal rewinfo.morph_rlt envs
(rewinfo.pattern , rewinfo.subject)
in
let goal, ir = Theory.Trans.ir_of_envs goal rewinfo.morph_rlt envs in
let units = Theory.Trans.ir_to_units ir in
let m = Matcher.subterm ?strict units pattern subject in
(* We sort the monad in increasing size of contet *)
let m = Search_monad.sort (fun (x,_,_) (y,_,_) -> x - y) m in
if show
then
Print.print rewinfo.morph_rlt ir m (hypinfo.Coq.Rewrite.context)
else
try
let pat,subst = choose_subst occ_subterm occ_sol m in
let tr_step = Matcher.Subst.instantiate subst pat in
let tr_step_raw = Theory.Trans.raw_constr_of_t ir rewinfo.morph_rlt [] tr_step in
let conv = (Theory.Trans.raw_constr_of_t ir rewinfo.morph_rlt (hypinfo.Coq.Rewrite.context)) in
let subst = Matcher.Subst.to_list subst in
let subst = List.map (fun (x,y) -> x, conv y) subst in
let by_aac_reflexivity = (by_aac_reflexivity rewinfo.subject rewinfo.eqt ir) in
let env = Tacmach.pf_env goal in
let sigma = Tacmach.project goal in
(* I'm not sure whether this is the right env/sigma for printing tr_step_raw *)
core_aac_rewrite ?abort rewinfo subst by_aac_reflexivity env sigma tr_step_raw tr_step subject
with
| NoSolutions ->
Tacticals.tclFAIL 0
(Pp.str (if occ_subterm = None && occ_sol = None
then "No matching occurrence modulo AC found"
else "No such solution"))
)
) goal
let get k l = try Some (List.assoc k l) with Not_found -> None
let get_lhs l = try ignore (List.assoc "in_right" l); false with Not_found -> true
let aac_rewrite ~args =
aac_rewrite_wrap ~occ_subterm:(get "at" args) ~occ_sol:(get "subst" args) ~in_left:(get_lhs args)
let rec add k x = function
| [] -> [k,x]
| k',_ as ky::q ->
if k'=k then Coq.user_error @@ Pp.strbrk ("redondant argument ("^k^")")
else ky::add k x q
let pr_aac_args _ _ _ l =
List.fold_left
(fun acc -> function
| ("in_right" as s,_) -> Pp.(++) (Pp.str s) acc
| (k,i) -> Pp.(++) (Pp.(++) (Pp.str k) (Pp.int i)) acc
) (Pp.str "") l
|